## Fachbereich Mathematik

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- Fachbereich Mathematik (227)
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Structure and Construction of Instanton Bundles on P3

The study of families of curves with prescribed singularities has a long tradition. Its foundations were laid by Plücker, Severi, Segre, and Zariski at the beginning of the 20th century. Leading to interesting results with applications in singularity theory and in the topology of complex algebraic curves and surfaces it has attained the continuous attraction of algebraic geometers since then. Throughout this thesis we examine the varieties V(D,S1,...,Sr) of irreducible reduced curves in a fixed linear system |D| on a smooth projective surface S over the complex numbers having precisely r singular points of types S1,...,Sr. We are mainly interested in the following three questions: 1) Is V(D,S1,...,Sr) non-empty? 2) Is V(D,S1,...,Sr) T-smooth, that is smooth of the expected dimension? 3) Is V(D,S1,...Sr) irreducible? We would like to answer the questions in such a way that we present numerical conditions depending on invariants of the divisor D and of the singularity types S1,...,Sr, which ensure a positive answer. The main conditions which we derive will be of the type inv(S1)+...+inv(Sr) < aD^2+bD.K+c, where inv is some invariant of singularity types, a, b and c are some constants, and K is some fixed divisor. The case that S is the projective plane has been very well studied by many authors, and on other surfaces some results for curves with nodes and cusps have been derived in the past. We, however, consider arbitrary singularity types, and the results which we derive apply to large classes of surfaces, including surfaces in projective three-space, K3-surfaces, products of curves and geometrically ruled surfaces.

Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.

Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)

Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.

In this work we present and estimate an explanatory model with a predefined system of explanatory equations, a so called lag dependent model. We present a locally optimal, on blocked neural network based lag estimator and theorems about consistensy. We define the change points in context of lag dependent model, and present a powerfull algorithm for change point detection in high dimensional high dynamical systems. We present a special kind of bootstrap for approximating the distribution of statistics of interest in dependent processes.

Different aspects of geomagnetic field modelling from satellite data are examined in the framework of modern multiscale approximation. The thesis is mostly concerned with wavelet techniques, i.e. multiscale methods based on certain classes of kernel functions which are able to realize a multiscale analysis of the funtion (data) space under consideration. It is thus possible to break up complicated functions like the geomagnetic field, electric current densities or geopotentials into different pieces and study these pieces separately. Based on a general approach to scalar and vectorial multiscale methods, topics include multiscale denoising, crustal field approximation and downward continuation, wavelet-parametrizations of the magnetic field in Mie-representation as well as multiscale-methods for the analysis of time-dependent spherical vector fields. For each subject the necessary theoretical framework is established and numerical applications examine and illustrate the practical aspects.

One crucial assumption of continuous financial mathematics is that the portfolio can be rebalanced continuously and that there are no transaction costs. In reality, this of course does not work. On the one hand, continuous rebalancing is impossible, on the other hand, each transaction causes costs which have to be subtracted from the wealth. Therefore, we focus on trading strategies which are based on discrete rebalancing - in random or equidistant times - and where transaction costs are considered. These strategies are considered for various utility functions and are compared with the optimal ones of continuous trading.

The immiscible lattice BGK method for solving the two-phase incompressible Navier-Stokes equations is analysed in great detail. Equivalent moment analysis and local differential geometry are applied to examine how interface motion is determined and how surface tension effects can be included such that consistency to the two-phase incompressible Navier-Stokes equations can be expected. The results obtained from theoretical analysis are verified by numerical experiments. Since the intrinsic interface tracking scheme of immiscible lattice BGK is found to produce unsatisfactory results in two-dimensional simulations several approaches to improving it are discussed but all of them turn out to yield no substantial improvement. Furthermore, the intrinsic interface tracking scheme of immiscible lattice BGK is found to be closely connected to the well-known conservative volume tracking method. This result suggests to couple the conservative volume tracking method for determining interface motion with the Navier-Stokes solver of immiscible lattice BGK. Applied to simple flow fields, this coupled method yields much better results than plain immiscible lattice BGK.

The dissertation is concerned with the numerical solution of Fokker-Planck equations in high dimensions arising in the study of dynamics of polymeric liquids. Traditional methods based on tensor product structure are not applicable in high dimensions for the number of nodes required to yield a fixed accuracy increases exponentially with the dimension; a phenomenon often referred to as the curse of dimension. Particle methods or finite point set methods are known to break the curse of dimension. The Monte Carlo method (MCM) applied to such problems are 1/sqrt(N) accurate, where N is the cardinality of the point set considered, independent of the dimension. Deterministic version of the Monte Carlo method called the quasi Monte Carlo method (QMC) are quite effective in integration problems and accuracy of the order of 1/N can be achieved, up to a logarithmic factor. However, such a replacement cannot be carried over to particle simulations due to the correlation among the quasi-random points. The method proposed by Lecot (C.Lecot and F.E.Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity, 15 (1999), pp.342-359) is the only known QMC approach, but it not only leads to large particle numbers but also the proven order of convergence is 1/N^(2s) in dimension s. We modify the method presented there, in such a way that the new method works with reasonable particle numbers even in high dimensions and has better order of convergence. Though the provable order of convergence is 1/sqrt(N), the results show less variance and thus the proposed method still slightly outperforms standard MCM.

In the present work, we investigated how to correct the questionable normality, linear and quadratic assumptions underlying existing Value-at-Risk methodologies. In order to take also into account the skewness, the heavy tailedness and the stochastic feature of the volatility of the market values of financial instruments, the constant volatility hypothesis widely used by existing Value-at-Risk appproches has also been investigated and corrected and the tails of the financial returns distributions have been handled via Generalized Pareto or Extreme Value Distributions. Artificial Neural Networks have been combined by Extreme Value Theory in order to build consistent and nonparametric Value-at-Risk measures without the need to make any of the questionable assumption specified above. For that, either autoregressive models (AR-GARCH) have been used or the direct characterization of conditional quantiles due to Bassett, Koenker [1978] and Smith [1987]. In order to build consistent and nonparametric Value-at-Risk estimates, we have proved some new results extending White Artificial Neural Network denseness results to unbounded random variables and provide a generalisation of the Bernstein inequality, which is needed to establish the consistency of our new Value-at-Risk estimates. For an accurate estimation of the quantile of the unexpected returns, Generalized Pareto and Extreme Value Distributions have been used. The new Artificial Neural Networks denseness results enable to build consistent, asymptotically normal and nonparametric estimates of conditional means and stochastic volatilities. The denseness results uses the Sobolev metric space L^m (my) for some m >= 1 and some probability measure my and which holds for a certain subclass of square integrable functions. The Fourier transform, the new extension of the Bernstein inequality for unbounded random variables from stationary alpha-mixing processes combined with the new generalization of a result of White and Wooldrige [1990] have been the main tool to establich the extension of White's neural network denseness results. To illustrate the goodness and level of accuracy of the new denseness results, we were able to demonstrate the applicability of the new Value-at-Risk approaches by means of three examples with real financial data mainly from the banking sector traded on the Frankfort Stock Exchange.

This thesis builds a bridge between singularity theory and computer algebra. To an isolated hypersurface singularity one can associate a regular meromorphic connection, the Gauß-Manin connection, containing a lattice, the Brieskorn lattice. The leading terms of the Brieskorn lattice with respect to the weight and V-filtration of the Gauß-Manin connection define the spectral pairs. They correspond to the Hodge numbers of the mixed Hodge structure on the cohomology of the Milnor fibre and belong to the finest known invariants of isolated hypersurface singularities. The differential structure of the Brieskorn lattice can be described by two complex endomorphisms A0 and A1 containing even more information than the spectral pairs. In this thesis, an algorithmic approach to the Brieskorn lattice in the Gauß-Manin connection is presented. It leads to algorithms to compute the complex monodromy, the spectral pairs, and the differential structure of the Brieskorn lattice. These algorithms are implemented in the computer algebra system Singular.

Diese Arbeit gehört in die algebraische Geometrie und die Darstellungstheorie und stellt eine Beziehung zwischen beiden Gebieten dar. Man beschäftigt sich mit den abgeleiteten Kategorien auf flachen Entartungen projektiver Geraden und elliptischer Kurven. Als Mittel benutzt man die Technik der Matrixprobleme. Das Hauptergebnis dieser Dissertation ist der folgende Satz: SATZ. Sei X ein Zykel projektiver Geraden. Dann gibt es drei Typen unzerlegbarer Objekte in D^-(Coh_X): - Shifts von Wolkenkratzergarben in einem regulären Punkt; - Bänder B(w,m,lambda), - Saiten S(w). Ganz analog beweist man die Zahmheit der abgeleiteten Kategorien vieler assoziativer Algebren.

The central theme in this thesis concerns the development of enhanced methods and algorithms for appraising market and credit risks and their application within the context of standard and more advanced market models. Generally, methods and algorithms for analysing market risk of complex portfolios involve detailed knowledge of option sensitivities, the so-called "Greeks". Based on an analysis of symmetries in financial market models, relations between option sensitivities are obtained, which can be used for the efficient valuation of the Greeks. Mainly, the relations are derived within the Black Scholes model, however, some relations are also valid for more general models, for instance the Heston model. Portfolios are usually influenced by lots of underlyings, so it is necessary to characterise the dependencies of these basic instruments. It is usual to describe such dependencies by correlation matrices. However, estimations of correlation matrices in practice are disturbed by statistical noise and usually have the problem of rank deficiency due to missing data. A fast algorithm is presented which performs a generalized Cholesky decomposition of a perturbed correlation matrix. In contrast to the standard Cholesky algorithm, an advantage of the generalized method is that it works for semi-positive, rank deficient matrices as well. Moreover, it gives an approximative decomposition when the input matrix is indefinite. A comparison with known algorithms with similar features is performed and it turns out, that the new algorithm can be recommended in situations where computation time is the critical issue. The determination of a profit and loss distribution by Fourier inversion of its characteristic function is a powerful tool, but it can break down when the characteristic function is not integrable. In this thesis, methods for Fourier inversion of non-integrable characteristic functions are studied. In this respect, two theorems are obtained which are based on a suitable approximation of the unknown distribution with known density and characteristic function. Further it will be shown, that straightforward Fast Fourier inversion works, when the according density lives on a bounded interval. The above techniques are of crucial importance to determine the profit and loss distribution (P&L) of large portfolios efficiently. The so-called Delta Gamma normal approach has become industrial standard for the estimation of market risk. It is shown, that the performance of the Delta Gamma normal approach can be improved substantially by application of the developed methods. The same optimization procedure also applies to the Delta Gamma Student model. A standard tool for computing the P&L distribution of a loan portfolio is the CreditRisk+ model. Basically, the CreditRisk+ distribution is a discrete distribution which can be computed from its probability generating function. For this a numerically stable method is presented and as an alternative, a new algorithm based on Fourier inversion is proposed. Finally, an extension of the CreditRisk+ model to market risk is developed, which distribution can be obtained efficiently by the presented Fourier inversion methods as well.

The thesis deals with the subgradient optimization methods which are serving to solve nonsmooth optimization problems. We are particularly concerned with solving large-scale integer programming problems using the methodology of Lagrangian relaxation and dualization. The goal is to employ the subgradient optimization techniques to solve large-scale optimization problems that originated from radiation therapy planning problem. In the thesis, different kinds of zigzagging phenomena which hamper the speed of the subgradient procedures have been investigated and identified. Moreover, we have established a new procedure which can completely eliminate the zigzagging phenomena of subgradient methods. Procedures used to construct both primal and dual solutions within the subgradient schemes have been also described. We applied the subgradient optimization methods to solve the problem of minimizing total treatment time of radiation therapy. The problem is NP-hard and thus far there exists no method for solving the problem to optimality. We present a new, efficient, and fast algorithm which combines exact and heuristic procedures to solve the problem.

The thesis is concerned with the modelling of ionospheric current systems and induced magnetic fields in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered operator equation. First of all a general multiscale concept for vectorial operator equations between two separable Hilbert spaces is developed in terms of vector kernel functions. The equivalence to the canonical tensorial ansatz is proven and the theory is transferred to the case of multiscale regularization of vectorial inverse problems. As a first application, a special multiresolution analysis of the space of square-integrable vector fields on the sphere, e.g. the Earth’s magnetic field measured on a spherical satellite’s orbit, is presented. By this, a multiscale separation of spherical vector-valued functions with respect to their sources can be established. The vector field is split up into a part induced by sources inside the sphere, a part which is due to sources outside the sphere and a part which is generated by sources on the sphere, i.e. currents crossing the sphere. The multiscale technqiue is tested on a magnetic field data set of the satellite CHAMP and it is shown that crustal field determination can be improved by previously applying our method. In order to reconstruct ionspheric current systems from magnetic field data, an inversion of the Biot-Savart’s law in terms of multiscale regularization is defined. The corresponding operator is formulated and the singular values are calculated. Based on the konwledge of the singular system a regularzation technique in terms of certain product kernels and correponding convolutions can be formed. The method is tested on different simulations and on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

We construct and study two surface measures on the space C([0,1],M) of paths in a compact Riemannian manifold M embedded into the Euclidean space R^n. The first one is induced by conditioning the usual Wiener measure on C([0,T],R^n) to the event that the Brownian particle does not leave the tubular epsilon-neighborhood of M up to time T, and passing to the limit. The second one is defined as the limit of the laws of reflected Brownian motions with reflection on the boundaries of the tubular epsilon-neighborhoods of M. We prove that the both surface measures exist and compare them with the Wiener measure W_M on C([0,T],M). We show that the first one is equivalent to W_M and compute the corresponding density explicitly in terms of the scalar curvature and the mean curvature vector of M. Further, we show that the second surface measure coincides with W_M. Finally, we study the limit behavior of the both surface measures as T tends to infinity.

In this thesis the combinatorial framework of toric geometry is extended to equivariant sheaves over toric varieties. The central questions are how to extract combinatorial information from the so developed description and whether equivariant sheaves can, like toric varieties, be considered as purely combinatorial objects. The thesis consists of three main parts. In the first part, by systematically extending the framework of toric geometry, a formalism is developed for describing equivariant sheaves by certain configurations of vector spaces. In the second part, homological properties of a certain class of equivariant sheaves are investigated, namely that of reflexive equivariant sheaves. Several kinds of resolutions for these sheaves are constructed which depend only on the configuration of their associated vector spaces. Thus a partially positive answer to the question of combinatorial representability is given. As a particular result, a new way for computing minimal resolutions for Z^n - graded modules over polynomial rings is obtained. In the third part a complete classification of the simplest nontrivial sheaves, equivariant vector bundles of rank two over smooth toric surfaces, is given. A combinatorial characterization is given and parameter spaces (moduli spaces) are constructed which depend only on this characterization. In appendices a outlook on equivariant sheaves and the relation of Chern classes to their combinatorial classification is given, particularly focussing on the case of the projective plane. A classification of equivariant vector bundles of rank three over the projective plane is given.

Semiparametric estimation of conditional quantiles for time series, with applications in finance
(2003)

The estimation of conditional quantiles has become an increasingly important issue in insurance and financial risk management. The stylized facts of financial time series data has rendered direct applications of extreme value theory methodologies, in the estimation of extreme conditional quantiles, inappropriate. On the other hand, quantile regression based procedures work well in nonextreme parts of a given data but breaks down in extreme probability levels. In order to solve this problem, we combine nonparametric regressions for time series and extreme value theory approaches in the estimation of extreme conditional quantiles for financial time series. To do so, a class of time series models that is similar to nonparametric AR-(G)ARCH models but which does not depend on distributional and moments assumptions, is introduced. We discuss estimation procedures for the nonextreme levels using the models and consider the estimates obtained by inverting conditional distribution estimators and by direct estimation using Koenker-Basset (1978) version for kernels. Under some regularity conditions, the asymptotic normality and uniform convergence, with rates, of the conditional quantile estimator for strong mixing time series, are established. We study the estimation of scale function in the introduced models using similar procedures and show that under some regularity conditions, the scale estimate is weakly consistent and asymptotically normal. The application of introduced models in the estimation of extreme conditional quantiles is achieved by augmenting them with methods in extreme value theory. It is shown that the overal extreme conditional quantiles estimator is consistent. A Monte Carlo study is carried out to illustrate the good performance of the estimates and real data are used to demonstrate the estimation of Value-at-Risk and conditional expected shortfall in financial risk management and their multiperiod predictions discussed.

Extensions of Shallow Water Equations The subject of the thesis of Michael Hilden is the simulation of floods in urban areas. In case of strong rain events, water can flow out of the overloaded sewer system onto the street and damage the connected houses. The dependable simulation of water flow out of a manhole ("manhole") and over a curb ("curb") is crucial for the assessment of the flood risks. The incompressible 3D-Navier-Stokes Equations (3D-NSE) describe the free surface flow of water accurately, but require expensive computations. Therefore, the less CPU-intensive (factor ca.1/100) Shallow Water Equations (SWE) are usually applied in hydrology. They can be derived from 3D-NSE under the assumption of a hydrostatic pressure distribution via depth-integration and are applied successfully in particular to simulations of river flow processes. The SWE-computations of the flow problems "manhole" and "curb" differ to the 3D-NSE results. Thus, SWE need to be extended appropriately to give reliable forecasts for flood risks in urban areas within reduced computational efforts. These extensions are developed based on physical considerations not considered in the classical SWE. In one extension, a vortex layer on the ground is separated from the main flow representing its new bottom. In a further extension, the hydrostatic pressure distribution is corrected by additional terms due to approximations of vertical velocities and their interaction with the flow. These extensions increase the quality of the SWE results for these flow problems up to the quality level of the NSE results within a moderate increase of the CPU efforts.

The thesis discusses discrete-time dynamic flows over a finite time horizon T. These flows take time, called travel time, to pass an arc of the network. Travel times, as well as other network attributes, such as, costs, arc and node capacities, and supply at the source node, can be constant or time-dependent. Here we review results on discrete-time dynamic flow problems (DTDNFP) with constant attributes and develop new algorithms to solve several DTDNFPs with time-dependent attributes. Several dynamic network flow problems are discussed: maximum dynamic flow, earliest arrival flow, and quickest flow problems. We generalize the hybrid capacity scaling and shortest augmenting path algorithmic of the static network flow problem to consider the time dependency of the network attributes. The result is used to solve the maximum dynamic flow problem with time-dependent travel times and capacities. We also develop a new algorithm to solve earliest arrival flow problems with the same assumptions on the network attributes. The possibility to wait (or park) at a node before departing on outgoing arc is also taken into account. We prove that the complexity of new algorithm is reduced when infinite waiting is considered. We also report the computational analysis of this algorithm. The results are then used to solve quickest flow problems. Additionally, we discuss time-dependent bicriteria shortest path problems. Here we generalize the classical shortest path problems in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a travel time on each arc. These problems have several interesting practical applications, but have not attained much attention in the literature. Here we develop two new algorithms in which one of them requires weaker assumptions as in previous research on the subject. Numerical tests show the superiority of the new algorithms. We then apply dynamic network flow models and their associated solution algorithms to determine lower bounds of the evacuation time, evacuation routes, and maximum capacities of inhabited areas with respect to safety requirements. As a macroscopic approach, our dynamic network flow models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building.

The main two problems of continuous-time financial mathematics are option pricing and portfolio optimization. In this thesis, various new aspects of these major topics of financial mathematics will be discussed. In all our considerations we will assume the standard diffusion type setting for securitiy prices which is today well-know under the term "Black-Scholes model". This setting and the basic results of option pricing and portfolio optimization are surveyed in the first chapter. The next three chapters deal with generalizations of the standard portfolio problem, also know as "Merton's problem". Here, we will always use the stochastic control approach as introduced in the seminal papers by Merton (1969, 1971, 1990). One such problem is the very realistic setting of an investor who is faced with fixed monetary streams. More precisely, in addition to maximizing the utility from final wealth via choosing an investment strategy, the investor also has to fulfill certain consumption needs. Also the opposite situation, an additional income stream can now be taken into account in our portfolio optimization problem. We consider various examples and solve them on one hand via classical stochastic control methods and on the other hand by our new separation theorem. This together with some numerical examples forms Chapter 2. Chapter 3 is mainly concerned with the portfolio problem if the investor has different lending and borrowing rates. We give explicit solutions (where possible) and numerical methods to calculate the optimal strategy in the cases of log utility and HARA utility for three different modelling approaches of the dependence of the borrowing rate on the fraction of wealth financed by a credit. The further generalization of the standard Merton problem in Chapter 4 consists in considering simultaneously the possibilities for continuous and discrete consumption. In our general approach there is a possibility for assigning the different consumption times different weights which is a generalization of the usual way of making them comparable via discounting. Chapter 5 deals with the special case of pricing basket options. Here, the main problem is not path-dependence but the multi-dimensionality which makes it impossible to give usuefull analytical representations of the option price. We review the literature and compare six different numerical methods in a systematic way. Thereby we also look at the influence of various parameters such as strike, correlation, forwards or volatilities on the erformance of the different numerical methods. The problem of pricing Asian options on average spot with average strike is the topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approximate option price. This method proves to be very reliable and e±cient for the valuation of different variants of Asian options on average spot with average strike.

The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.

The present thesis deals with coupled steady state laminar flows of isothermal incompressible viscous Newtonian fluids in plain and in porous media. The flow in the pure fluid region is usually described by the (Navier-)Stokes system of equations. The most popular models for the flow in the porous media are those suggested by Darcy and by Brinkman. Interface conditions, proposed in the mathematical literature for coupling Darcy and Navier-Stokes equations, are shortly reviewed in the thesis. The coupling of Navier-Stokes and Brinkman equations in the literature is based on the so called continuous stress tensor interface conditions. One of the main tasks of this thesis is to investigate another type of interface conditions, namely, the recently suggested stress tensor jump interface conditions. The mathematical models based on these interface conditions were not carefully investigated from the mathematical point of view, and also their validity was a subject of discussions. The considerations within this thesis are a step toward better understanding of these interface conditions. Several aspects of the numerical simulations of such coupled flows are considered: -the choice of proper interface conditions between the plain and porous media -analysis of the well-posedness of the arising systems of partial differential equations; -developing numerical algorithm for the stress tensor jump interface conditions, coupling Navier-Stokes equations in the pure liquid media with the Navier-Stokes-Brinkman equations in the porous media; -validation of the macroscale mathematical models on the base of a comparison with the results from a direct numerical simulation of model representative problems, allowing for grid resolution of the pore level geometry; -developing software and performing numerical simulation of 3-D industrial flows, namely of oil flows through car filters.

The question of how to model dependence structures between financial assets was revolutionized since the last decade when the copula concept was introduced in financial research. Even though the concept of splitting marginal behavior and dependence structure (described by a copula) of multidimensional distributions already goes back to Sklar (1955) and Hoeffding (1940), there were very little empirical efforts done to check out the potentials of this approach. The aim of this thesis is to figure out the possibilities of copulas for modelling, estimating and validating purposes. Therefore we extend the class of Archimedean Copulas via a transformation rule to new classes and come up with an explicit suggestion covering the Frank and Gumbel family. We introduce a copula based mapping rule leading to joint independence and as results of this mapping we present an easy method of multidimensional chi²-testing and a new estimate for high dimensional parametric distributions functions. Different ways of estimating the tail dependence coefficient, describing the asymptotic probability of joint extremes, are compared and improved. The limitations of elliptical distributions are carried out and a generalized form of them, preserving their applicability, is developed. We state a method to split a (generalized) elliptical distribution into its radial and angular part. This leads to a positive definite robust estimate of the dispersion matrix (here only given as a theoretical outlook). The impact of our findings is stated by modelling and testing the return distributions of stock- and currency portfolios furthermore of oil related commodities- and LME metal baskets. In addition we show the crash stability of real estate based firms and the existence of nonlinear dependence in between the yield curve.

Nowadays one of the major objectives in geosciences is the determination of the gravitational field of our planet, the Earth. A precise knowledge of this quantity is not just interesting on its own but it is indeed a key point for a vast number of applications. The important question is how to obtain a good model for the gravitational field on a global scale. The only applicable solution - both in costs and data coverage - is the usage of satellite data. We concentrate on highly precise measurements which will be obtained by GOCE (Gravity Field and Steady State Ocean Circulation Explorer, launch expected 2006). This satellite has a gradiometer onboard which returns the second derivatives of the gravitational potential. Mathematically seen we have to deal with several obstacles. The first one is that the noise in the different components of these second derivatives differs over several orders of magnitude, i.e. a straightforward solution of this outer boundary value problem will not work properly. Furthermore we are not interested in the data at satellite height but we want to know the field at the Earth's surface, thus we need a regularization (downward-continuation) of the data. These two problems are tackled in the thesis and are now described briefly. Split Operators: We have to solve an outer boundary value problem at the height of the satellite track. Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We present a different approach: We classify all first and purely second order operators which fulfill that a harmonic function stays harmonic under their application. This task is done by using modern algebraic methods for solving systems of partial differential equations symbolically. Now we can look at the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of the differential operator, i.e. integration. In particular we are capable to deal with derivatives which are tangential to the boundary. Auto-Regularization: The second obstacle is finding a proper regularization procedure. This is complicated by the fact that we are facing stochastic rather than deterministic noise. The main question is how to find an optimal regularization parameter which is impossible without any additional knowledge. However we could show that with a very limited number of additional information, which are obtainable also in practice, we can regularize in an asymptotically optimal way. In particular we showed that the knowledge of two input data sets allows an order optimal regularization procedure even under the hard conditions of Gaussian white noise and an exponentially ill-posed problem. A last but rather simple task is combining data from different derivatives which can be done by a weighted least squares approach using the information we obtained out of the regularization procedure. A practical application to the downward-continuation problem for simulated gravitational data is shown.

In traditional portfolio optimization under the threat of a crash the investment horizon or time to maturity is neglected. Developing the so-called crash hedging strategies (which are portfolio strategies which make an investor indifferent to the occurrence of an uncertain (down) jumps of the price of the risky asset) the time to maturity turns out to be essential. The crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy. Hereby the situation of changing market coefficients after a possible crash is considered for the case of logarithmic utility as well as for the case of general utility functions. A benefit-cost analysis of the crash hedging strategy is done as well as a comparison of the crash hedging strategy with the optimal portfolio strategies given in traditional crash models. Moreover, it will be shown that the crash hedging strategies optimize the worst-case bound for the expected utility from final wealth subject to some restrictions. Another application is to model crash hedging strategies in situations where both the number and the height of the crash are uncertain but bounded. Taking the additional information of the probability of a possible crash happening into account leads to the development of the q-quantile crash hedging strategy.

In the filling process of a car tank, the formation of foam plays an unwanted role, as it may prevent the tank from being completely filled or at least delay the filling. Therefore it is of interest to optimize the geometry of the tank using numerical simulation in such a way that the influence of the foam is minimized. In this dissertation, we analyze the behaviour of the foam mathematically on the mezoscopic scale, that is for single lamellae. The most important goals are on the one hand to gain a deeper understanding of the interaction of the relevant physical effects, on the other hand to obtain a model for the simulation of the decay of a lamella which can be integrated in a global foam model. In the first part of this work, we give a short introduction into the physical properties of foam and find that the Marangoni effect is the main cause for its stability. We then develop a mathematical model for the simulation of the dynamical behaviour of a lamella based on an asymptotic analysis using the special geometry of the lamella. The result is a system of nonlinear partial differential equations (PDE) of third order in two spatial and one time dimension. In the second part, we analyze this system mathematically and prove an existence and uniqueness result for a simplified case. For some special parameter domains the system can be further simplified, and in some cases explicit solutions can be derived. In the last part of the dissertation, we solve the system using a finite element approach and discuss the results in detail.

In this text we survey some large deviation results for diffusion processes. The first chapters present results from the literature such as the Freidlin-Wentzell theorem for diffusions with small noise. We use these results to prove a new large deviation theorem about diffusion processes with strong drift. This is the main result of the thesis. In the later chapters we give another application of large deviation results, namely to determine the exponential decay rate for the Bayes risk when separating two different processes. The final chapter presents techniques which help to experiment with rare events for diffusion processes by means of computer simulations.

In this dissertation we consider complex, projective hypersurfaces with many isolated singularities. The leading questions concern the maximal number of prescribed singularities of such hypersurfaces in a given linear system, and geometric properties of the equisingular stratum. In the first part a systematic introduction to the theory of equianalytic families of hypersurfaces is given. Furthermore, the patchworking method for constructing hypersurfaces with singularities of prescribed types is described. In the second part we present new existence results for hypersurfaces with many singularities. Using the patchworking method, we show asymptotically proper results for hypersurfaces in P^n with singularities of corank less than two. In the case of simple singularities, the results are even asymptotically optimal. These statements improve all previous general existence results for hypersurfaces with these singularities. Moreover, the results are also transferred to hypersurfaces defined over the real numbers. The last part of the dissertation deals with the Castelnuovo function for studying the cohomology of ideal sheaves of zero-dimensional schemes. Parts of the theory of this function for schemes in P^2 are generalized to the case of schemes on general surfaces in P^3. As an application we show an H^1-vanishing theorem for such schemes.

In this thesis we show that the theory of algebraic correspondences introduced by Deuring in the 1930s can be applied to construct non-trivial homomorphisms between the Jacobi groups of hyperelliptic function fields. Concretely, we deduce algorithms to add and multiply correspondences which perform in a reasonable time if the degrees of the associated divisors of the double field are small. Moreover, we show how to compute the differential matrices associated to prime divisors of the double field for arbitrary genus. These matrices give a representation for the homomorphisms or endomorphisms in the additive group (ring) of matrices which is even faithful if the ground field has characteristic zero. As first examples for non-trivial correspondences we investigate multiplication by m endomorphisms. Afterwards we use factorisations of certain bivariate polynomials to construct prime divisors of the double field that are not equivalent to 0 in a coarser sense. Applying the theory of Deuring, these divisors yield homomorphisms between the Jacobi groups of special classes of hyperelliptic function fields. Finally, we generalise the Richelot isogeny to higher genus and by this way derive a class of hyperelliptic function fields given in terms of their defining polynomials which admit non-trivial homomorphisms. These include homomorphisms between the Jacobi groups of hyperelliptic curves of different as well as of equal genus. In addition we provide an explicit method to construct genus 2 function fields the endomorphism ring of which contains a sqrt(2) multiplication with the help of the Cholesky decomposition of a certain matrix.

In this thesis we propose an efficient method to compute the automorphism group of an arbitrary hyperelliptic function field over a given constant field of odd characteristic as well as over its algebraic extensions. Beside theoretical applications, knowing the automorphism group also is useful in cryptography: The Jacobians of hyperelliptic curves have been suggested by Koblitz as groups for cryptographic purposes, because the discrete logarithm is believed to be hard in this kind of groups. In order to obtain "secure" Jacobians, it is necessary to prevent attacks like Pohlig/Hellman's and Duursma/Gaudry/Morain's. The latter is only feasible, if the corresponding function field has an automorphism of large order. According to a theorem by Madan, automorphisms seem to allow the Pohlig/Hellman attack, too. Hence, the function field of a secure Jacobian will most likely have trivial automorphism group. In other words: Computing the automorphism group of a hyperelliptic function field promises to be a quick test for insecure Jacobians. Let us outline our algorithm for computing the automorphism group Aut(F/k) of a hyperelliptic function field F/k. It is well known that Aut(F/k) is finite. For each possible subgroup U of Aut(F/k), Rolf Brandt has given a normal form for F if k is algebraically closed. Hence our problem reduces to deciding, whether a given hyperelliptic function field F=k(x,y), y^2=D_x has a defining equation of the form given by Brandt. This question can be answered using theorem III.18: We have F=k(t,u), u^2=D_t iff x is a fraction of linear polynomials in t and y=pu, where the factor p is a rational function w.r.t. t which can be determined explicitly from the coefficients of x. This condition can be checked efficiently using Gröbner basis techniques. With additional effort, it is also possible to compute Aut(F/k) if k is not algebraically closed. Investigating a huge number of examples one gets the impression that the above motivation of getting a quick test for insecure Jacobians is partially fulfilled: The computation of automorphism groups is quite fast using the suggested algorithm. Furthermore, fields with nontrivial automorphism groups seem to have insecure Jacobians. Only fields of small characteristic seem to have a reasonable chance of having nontrivial automorphisms. Hence, from a cryptographic point of view, computing Aut(F/k) seems to make sense whenever k has small characteristic.

In this dissertation a model of melt spinning (by Doufas, McHugh and Miller) has been investigated. The model (DMM model) which takes into account effects of inertia, air drag, gravity and surface tension in the momentum equation and heat exchange between air and fibre surface, viscous dissipation and crystallization in the energy equation also has a complicated coupling with the microstructure. The model has two parts, before onset of crystallization (BOC) and after onset of crystallization (AOC) with the point of onset of crystallization as the unknown interface. Mathematically the model has been formulated as a Free boundary value problem. Changes have been introduced in the model with respect to the air drag and an interface condition at the free boundary. The mathematical analysis of the nonlinear, coupled free boundary value problem shows that the solution of this problem depends heavily on initial conditions and parameters which renders the global analysis impossible. But by defining a physically acceptable solution, it is shown that for a more restricted set of initial conditions if a unique solution exists for IVP BOC then it is physically acceptable. For this the important property of the positivity of the conformation tensor variables has been proved. Further it is shown that if a physically acceptable solution exists for IVP BOC then under certain conditions it also exists for IVP AOC. This gives an important relation between the initial conditions of IVP BOC and the existence of a physically acceptable solution of IVP AOC. A new investigation has been done for the melt spinning process in the framework of classical mechanics. A Hamiltonian formulation has been done for the melt spinning process for which appropriate Poisson brackets have been derived for the 1-d, elongational flow of a viscoelastic fluid. From the Hamiltonian, cross sectionally averaged balance mass and momentum equations of melt spinning can be derived along with the microstructural equations. These studies show that the complicated problem of melt spinning can also be studied under the framework of classical mechanics. This work provides the basic groundwork on which further investigations on the dynamics of a fibre could be carried out. The Free boundary value problem has been solved numerically using shooting method. Matlab routines have been used to solve the IVPs arising in the problem. Some numerical case studies have been done to study the sensitivity of the ODE systems with respect to the initial guess and parameters. These experiments support the analysis done and throw more light on the stiff nature and ill posedness of the ODE systems. To validate the model, simulations have been performed on sets of data provided by the company. Comparison of numerical results (axial velocity profiles) has been done with the experimental profiles provided by the company. Numerical results have been found to be in excellent agreement with the experimental profiles.

This thesis investigates the constrained form of the spherical Minimax location problem and the spherical Weber location problem. Specifically, we consider the problem of locating a new facility on the surface of the unit sphere in the presence of convex spherical polygonal restricted regions and forbidden regions such that the maximum weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized and the sum of the weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized. It is assumed that a forbidden region is an area on the surface of the unit sphere where travel and facility location are not permitted and that distance is measured using the great circle arc distance. We represent a polynomial time algorithm for the spherical Minimax location problem for the special case where all the existing facilities are located on the surface of a hemisphere. Further, we have developed algorithms for spherical Weber location problem using barrier distance on a hemisphere as well as on the unit sphere.

Competing Neural Networks as Models for Non Stationary Financial Time Series -Changepoint Analysis-
(2005)

The problem of structural changes (variations) play a central role in many scientific fields. One of the most current debates is about climatic changes. Further, politicians, environmentalists, scientists, etc. are involved in this debate and almost everyone is concerned with the consequences of climatic changes. However, in this thesis we will not move into the latter direction, i.e. the study of climatic changes. Instead, we consider models for analyzing changes in the dynamics of observed time series assuming these changes are driven by a non-observable stochastic process. To this end, we consider a first order stationary Markov Chain as hidden process and define the Generalized Mixture of AR-ARCH model(GMAR-ARCH) which is an extension of the classical ARCH model to suit to model with dynamical changes. For this model we provide sufficient conditions that ensure its geometric ergodic property. Further, we define a conditional likelihood given the hidden process and a pseudo conditional likelihood in turn. For the pseudo conditional likelihood we assume that at each time instant the autoregressive and volatility functions can be suitably approximated by given Feedfoward Networks. Under this setting the consistency of the parameter estimates is derived and versions of the well-known Expectation Maximization algorithm and Viterbi Algorithm are designed to solve the problem numerically. Moreover, considering the volatility functions to be constants, we establish the consistency of the autoregressive functions estimates given some parametric classes of functions in general and some classes of single layer Feedfoward Networks in particular. Beside this hidden Markov Driven model, we define as alternative a Weighted Least Squares for estimating the time of change and the autoregressive functions. For the latter formulation, we consider a mixture of independent nonlinear autoregressive processes and assume once more that the autoregressive functions can be approximated by given single layer Feedfoward Networks. We derive the consistency and asymptotic normality of the parameter estimates. Further, we prove the convergence of Backpropagation for this setting under some regularity assumptions. Last but not least, we consider a Mixture of Nonlinear autoregressive processes with only one abrupt unknown changepoint and design a statistical test that can validate such changes.

Die vorliegende Arbeit wurde angeregt durch die in A.N. Borodin(2000) [Version of the Feynman-Kac Formula. Journal of Mathematical Sciences, 99(2):1044-1052, 2000] und in B. Simon(2000) [A Feynman-Kac Formula for Unbounded Semigroups. Canadian Math. Soc. Conf. Proc., 28:317-321, 2000] dargestellten Feynman-Kac-Formeln. Sie beschäftigt sich mit dem Problem, den Geltungsbereich der Feynman-Kac-Formel im Hinblick auf die Bedingungen der Potentiale und der Anfangsbedingung der zugehörigen partiellen Differentialgleichung zu erweitern. Es ist bekannt, dass die Feynman-Kac-Formel für beschränkte Potentiale gilt. Ausserdem gilt sie auch für Anfangsbedingungen, die im Raum \(C_{0}(\mathbb{R}^{n})\) oder im Raum \(C_{c}^{2}(\mathbb{R}^{n})\) liegen. Die Darstellung der Feynman-Kac-Formel für die Anfangsbedingung, die im Raum \(C_{c}^{2}(\mathbb{R}^{n})\) liegt, liefert die Lösung der partiellen Differentialgleichung. Wir können sie auch als stark stetige Halbgruppe auf dem Raum \(C_{0}(\mathbb{R}^{n})\) auffassen. Diese zwei verschiedenen Darstellungen sind äquivalent. In dieser Arbeit zeigen wir zunächst, dass die Feynman-Kac-Formel auch für unbeschränkte Potentiale \(V\) gilt, wobei \(|V(x)| \leq \varepsilon ||x||^{2} + C_{\varepsilon} \) für alle \(\varepsilon > 0; C_{\varepsilon} > 0\) und \(x \in \mathbb{R}^{n}\) ist. Ausserdem zeigen wir, dass sie für alle Anfangsbedingungen \(f\) gilt mit \(x \mapsto e^{-\varepsilon |x|^{2}} f(x) \in H^{2,2}(\mathbb{R}^{n})\). Der Beweis ist wahrscheinlichkeitstheoretisch und benutzt keine Spektraltheorie. Der spektraltheoretische Zugang, in dem eine Darstellung des Operators \(e^{-tH}\), wobei \(H = -\frac{1}{2} \Delta + V\) gegeben wird, wurde von B. Simon(2000) auch auf die obige Klasse von Potentialen ausgeweitet. Wir lassen zusätzlich auch Potentiale der Form \(V = V_{1} + V_{2}\) zu, wobei \(V_{1} \in L^{2}(\mathbb{R}^{3})\) ist und für alle \(\varepsilon > 0\) gibt es \(C_{\varepsilon} > 0\), so dass \(|V_{2}(x)| \leq\varepsilon ||x||^{2} + C_{\varepsilon}\) für alle \(x \in \mathbb{R}^{3}\) ist. Im Gegensatz zur klassischen Situation ist \(e^{-tH}\) jetzt ein unbeschränkter Operator. Schließlich wird in dieser Arbeit auch der Zusammenhang zwischen der Feynman-Kac-It\(\hat{o}\)-Formel, der Feynman-Kac-Formel und der Kolmogorov-Rückwärtsgleichung untersucht.

In many industrial applications fast and accurate solutions of linear elliptic partial differential equations are needed as one of the building blocks of more complex problems. The domains are often highly complex and meshing turns out to be expensive and difficult to obtain with a sufficient quality. In such cases methods with a regular, not boundary adapted grid offer an attractive alternative. The Explicit Jump Immersed Interface Method is one of these algorithms. The main interest of this work lies in solving the linear elasticity equations. For this purpose the existing EJIIM algorithm has been extended to three dimensions. The Poisson equation is always considered in parallel as the most typical representative of elliptic PDEs. During the work it became clear that EJIIM can have very high computational memory requirements. To overcome this problem an improvement, Reduced EJIIM is proposed. The main theoretical result in this work is the proof of the smoothing property of inverses of elliptic finite difference operators in two and three space dimensions. It is an often observed phenomena that the local truncation error is allowed to be of lower order along some lower dimensional manifold without influencing the global convergence order of the solution.

An autoregressive-ARCH model with possible exogeneous variables is treated. We estimate the conditional volatility of the model by applying feedforward networks to the residuals and prove consistency and asymptotic normality for the estimates under the rate of feedforward networks complexity. Recurrent neural networks estimates of GARCH and value-at-risk is studied. We prove consistency and asymptotic normality for the recurrent neural networks ARMA estimator under the rate of recurrent networks complexity. We also overcome the estimation problem in stochastic variance models in discrete time by feedforward networks and the introduction of a new distributions on the innovations. We use the method to calculate market risk such as expected shortfall and Value-at risk. We tested this distribution together with other new distributions on the GARCH family models against other common distributions on the financial market such as Normal Inverse Gaussian, normal and the Student's t- distributions. As an application of the models, some German stocks are studied and the different approaches are compared together with the most common method of GARCH(1,1) fit.

It is considered an analytical model of defaultable bond portfolio in terms of its face value process. The face value process dynamically evolves with time and incorporates changes caused by recovery payment on default followed by purchasing of new bonds. The further studies involve properties, distribution and control of the face value process.

This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.

In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.

Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.

In modern textile manufacturing industries, the function of human eyes to detect disturbances in the production processes which yield defective products is switched to cameras. The camera images are analyzed with various methods to detect these disturbances automatically. There are, however, still problems with in particular semi-regular textures which are typical for weaving patterns. We study three parts of that problem of automatic texture analysis: image smoothing, texture synthesis and defect detection. In image smoothing, we develop a two dimensional kernel smoothing method with locally and directionally adaptive bandwidths allowing correlation in the errors. Two approaches are used in synthesising texture. The first is based on constructing a generalized Ising energy function in the Markov Random Field setup, and for the second, we use two-dimensional periodic bootstrap methods for semi-regular texture synthesis. We treat defect detection as multihypothesis testing problem with the null hypothesis representing the absence of defects and the other hypotheses representing various types of defects. We develop a test based on a nonparametric regression setup, and we use the bootstrap for approximating the distribution of our test statistic.

Since its invention by Sir Allistair Pilkington in 1952, the float glass process has been used to manufacture long thin flat sheets of glass. Today, float glass is very popular due to its high quality and relatively low production costs. When producing thinner glass the main concern is to retain its optical quality, which can be deteriorated during the manufacturing process. The most important stage of this process is the floating part, hence is considered to be responsible for the loss in the optical quality. A series of investigations performed on the finite products showed the existence of many short wave patterns, which strongly affect the optical quality of the glass. Our work is concerned with finding the mechanism for wave development, taking into account all possible factors. In this thesis, we model the floating part of the process by an theoretical study of the stability of two superposed fluids confined between two infinite plates and subjected to a large horizontal temperature gradient. Our approach is to take into account the mixed convection effects (viscous shear and buoyancy), neglecting on the other hand the thermo-capillarity effects due to the length of our domain and the presence of a small stabilizing vertical temperature gradient. Both fluids are treated as Newtonian with constant viscosity. They are immiscible, incompressible, have very different properties and have a free surface between them. The lower fluid is a liquid metal with a very small kinematic viscosity, whereas the upper fluid is less dense. The two fluids move with different velocities: the speed of the upper fluid is imposed, whereas the lower fluid moves as a result of buoyancy effects. We examine the problem by means of small perturbation analysis, and obtain a system of two Orr-Sommerfeld equations coupled with two energy equations, and general interface and boundary conditions. We solve the system analytically in the long- and short- wave limit, by using asymptotic expansions with respect to the wave number. Moreover, we write the system in the form of a general eigenvalue problem and we solve the system numerically by using Chebyshev spectral methods for fluid dynamics. The results (both analytical and numerical) show the existence of the small-amplitude travelling waves, which move with constant velocity for wave numbers in the intermediate range. We show that the stability of the system is ensured in the long wave limit, a fact which is in agreement with the real float glass process. We analyze the stability for a wide range of wave numbers, Reynolds, Weber and Grashof number, and explain the physical implications on the dynamics of the problem. The consequences of the linear stability results are discussed. In reality in the float glass process, the temperature strongly influences the viscosity of both molten metal and hot glass, which will have direct consequences on the stability of the system. We investigate the linear stability of two superposed fluids with temperature dependent viscosities by considering a different model for the viscosity dependence of each fluid. Although, the temperature-viscosity relationships for glass and metal are more complex than those used in our computations, our intention is to emphasize the effects of this dependence on the stability of the system. It is known from the literature that in the case of one fluid, the heat, which causes viscosity to decrease along the domain, usually destabilizes the flow. For the two superposed fluids problem we investigate this behaviour and discuss the consequences of the linear stability in this new case.

Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many important algebras in the classes of \(G\)- and \(GR\)-algebras and studied their ring-theoretic properties. Let \(A\) be a \(G\)-algebra in \(n\) variables. We establish necessary and sufficient conditions for \(A\) to have a Poincar'e-Birkhoff-Witt (PBW) basis. Further on, we show that besides the existence of a PBW basis, \(A\) shares some other properties with the commutative polynomial ring \(\mathbb{K}[x_1,\ldots,x_n]\). In particular, \(A\) is a Noetherian integral domain of Gel'fand-Kirillov dimension \(n\). Both Krull and global homological dimension of \(A\) are bounded by \(n\); we provide examples of \(G\)-algebras where these inequalities are strict. Finally, we prove that \(A\) is Auslander-regular and a Cohen-Macaulay algebra. In order to perform symbolic computations with modules over \(GR\)-algebras, we generalize Gröbner bases theory, develop and respectively enhance new and existing algorithms. We unite the most fundamental algorithms in a suite of applications, called "Gröbner basics" in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two-sided Gröbner bases for bimodules, annihilators of left modules and operations with opposite algebras. An important role in Representation Theory is played by various subalgebras, like the center and the Gel'fand-Zetlin subalgebra. We discuss their properties and their relations to Gröbner bases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between \(GR\)-algebras. We provide new results and algorithms for computing the preimage of a left ideal under a morphism of \(GR\)-algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a module into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one-dimensional representations of a \(G\)-algebra \(A\), and prove, moreover, that if the set of finite dimensional representations of \(A\) over a ground field \(K\) is not empty, then the homological dimension of \(A\) equals \(n\). All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of computations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of algorithms; the comparison of timings shows that our implementation outperforms all of the modern systems with the combination of both broad functionality and fast implementation. In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time.

We work in the setting of time series of financial returns. Our starting point are the GARCH models, which are very common in practice. We introduce the possibility of having crashes in such GARCH models. A crash will be modeled by drawing innovations from a distribution with much mass on extremely negative events, while in ''normal'' times the innovations will be drawn from a normal distribution. The probability of a crash is modeled to be time dependent, depending on the past of the observed time series and/or exogenous variables. The aim is a splitting of risk into ''normal'' risk coming mainly from the GARCH dynamic and extreme event risk coming from the modeled crashes. We will present several incarnations of this modeling idea and give some basic properties like the conditional first and second moments. For the special case that we just have an ARCH dynamic we can establish geometric ergodicity and, thus, stationarity and mixing conditions. Also in the ARCH case we formulate (quasi) maximum likelihood estimators and can derive conditions for consistency and asymptotic normality of the parameter estimates. In a special case of genuine GARCH dynamic we are able to establish L_1-approximability and hence laws of large numbers for the processes itself. We can formulate a conditional maximum likelihood estimator in this case, but cannot completely establish consistency for them. On the practical side we look for the outcome of estimating models with genuine GARCH dynamic and compare the result to classical GARCH models. We apply the models to Value at Risk estimation and see that in comparison to the classical models many of ours seem to work better although we chose the crash distributions quite heuristically.

The aim of the thesis is the numerical investigation of saturated, stationary, incompressible Newtonian flow in porous media when inertia is not negligible. We focus our attention to the Navier-Stokes system with two pressures derived by two-scale homogenization. The thesis is subdivided into five Chapters. After the introductory remarks on porous media, filtration laws and upscaling methods, the first chapter is closed by stating the basic terminology and mathematical fundamentals. In Chapter 2, we start by formulating the Navier-Stokes equations on a periodic porous medium. By two-scale expansions of the velocity and pressure, we formally derive the Navier-Stokes system with two pressures. For the sake of completeness, known existence and uniqueness results are repeated and a convergence proof is given. Finally, we consider Stokes and Navier-Stokes systems with two pressures with respect to their relation to Darcy's law. Chapter 3 and Chapter 4 are devoted to the numerical solution of the nonlinear two pressure system. Therefore, we follow two approaches. The first approach which is developed in Chapter 3 is based on a splitting of the Navier-Stokes system with two pressures into micro and macro problems. The splitting is achieved by Taylor expanding the permeability function or by discretely computing the permeability function. The problems to be solved are a series of Stokes and Navier-Stokes problems on the periodicity cell. The Stokes problems are solved by an Uzawa conjugate gradient method. The Navier-Stokes equations are linearized by a least-squares conjugate gradient method, which leads to the solution of a sequence of Stokes problems. The macro problem consists of solving a nonlinear uniformly elliptic equation of second order. The least-squares linearization is applied to the macro problem leading to a sequence of Poisson problems. All equations will be discretized by finite elements. Numerical results are presented at the end of Chapter 3. The second approach presented in Chapter 4 relies on the variational formulation in a certain Hilbert space setting of the Navier-Stokes system with two pressures. The nonlinear problem is again linearized by the least-squares conjugate gradient method. We obtain a sequence of Stokes systems with two pressures. For the latter systems, we propose a fast solution method which relies on pre-computing Stokes systems on the periodicity cell for finite element basis functions acting as right hand sides. Finally, numerical results are discussed. In Chapter 5 we are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and the chapter is closed by a numerical investigation of the model with special focus on inertia effects.

In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.

Tropical geometry is a rather new field of algebraic geometry. The main idea is to replace algebraic varieties by certain piece-wise linear objects in R^n, which can be studied with the aid of combinatorics. There is hope that many algebraically difficult operations become easier in the tropical setting, as the structure of the objects seems to be simpler. In particular, tropical geometry shows promise for application in enumerative geometry. Enumerative geometry deals with the counting of geometric objects that are determined by certain incidence conditions. Until around 1990, not many enumerative questions had been answered and there was not much prospect of solving more. But then Kontsevich introduced the moduli space of stable maps which turned out to be a very useful concept for the study of enumerative geometry. A well-known problem of enumerative geometry is to determine the numbers N_cplx(d,g) of complex genus g plane curves of degree d passing through 3d+g-1 points in general position. Mikhalkin has defined the analogous number N_trop(d,g) for tropical curves and shown that these two numbers coincide (Mikhalkin's Correspondence Theorem). Tropical geometry supplies many new ideas and concepts that could be helpful to answer enumerative problems. However, as a rather new field, tropical geometry has to be studied more thoroughly. This thesis is concerned with the ``translation'' of well-known facts of enumerative geometry to tropical geometry. More precisely, the main results of this thesis are: - a tropical proof of the invariance of N_trop(d,g) of the position of the 3d+g-1 points, - a tropical proof for Kontsevich's recursive formula to compute N_trop(d,0) and - a tropical proof of Caporaso's and Harris' algorithm to compute N_trop(d,g). All results were derived in joint work with my advisor Andreas Gathmann. (Note that tropical research is not restricted to the translation of classically well-known facts, there are actually new results shown by means of tropical geometry that have not been known before. For example, Mikhalkin gave a tropical algorithm to compute the Welschinger invariant for real curves. This shows that tropical geometry can indeed be a tool for a better understanding of classical geometry.)

This thesis deals with modeling aspects of generalized Newtonian and of non-Newtonian fluids, as well as with development and validation of algorithms used in simulation of such fluids. The main contribution in the modeling part are the introduction and analysis of a new model for the generalized Newtonian fluids, where constitutive equation is of an algebraic form. Distinction between shear and extensional viscosities leads to anisotropic viscosity model. It can be considered as a natural extension of the well known (isotropic viscosity) Carreau model, which deals only with shear viscosity properties of the fluid. The proposed model takes additionally into account extensional viscosity properties. Numerical results show that the anisotropic viscosity model gives much better agreement with experimental observations than the isotropic one. Another contribution of the thesis consists of the development and analysis of robust and reliable algorithms for simulation of generalized Newtonian fluids. For such fluids the momentum equations are strongly coupled through mixed derivatives appearing in the viscous term (unlike the case of Newtonian fluids). It is shown in this thesis, that a careful treatment of those derivatives is essential in deriving robust algorithms. A modification of a standard SIMPLE-like algorithm is given, where all the viscous terms from the momentum equations are discretized in an implicit manner. Moreover, it is shown that a block diagonal preconditioner to the viscous operator is good enough to be used in simulations. Furthermore, different solution techniques, namely projection type methods (consists of solving momentum equations and pressure correction equation) and fully coupled methods (momentum and continuity equations are solved together), are compared. It is shown, that explicit discretization of the mixed derivatives lead to stability problems. Further, analytical estimates of eigenvalue distribution for three different preconditioners, applied to the transformed system arising after discretization and linearization of the momentum and continuity equations, are provided. We propose to apply a block Gauss-Seidel preconditioner to the transformed system. The analysis shows, that this preconditioner is able to cluster eigenvalues around unity independent of the transformation step. It is not the case for other preconditioners applied to the transformed system as discussed in the thesis. The block Gauss-Seidel preconditioner has also shown the best behavior (among all preconditioners discussed in the thesis) in numerical experiments. Further contribution consists of comparison and validation of numerical algorithms applied in simulations of non-Newtonian fluids modeled by time integral constitutive equations. Numerical results from simulations of dilute polymer solutions, described by the integral Oldroyd B model, have shown very good quantitative agreement with the results obtained by differential Oldroyd B counterpart in 4:1 planar contraction domain at low Weissenberg numbers. In this case, the Weissenberg number is changed by changing the relaxation time. However, contrary to the differential Oldroyd B model, the integral one allows to perform stable simulations also in the range of high Weissenberg numbers. Moreover, very good agreement with experimental observations has been achieved. Simulations of concentrated polymer solutions (polystyrene and polybutadiene solutions), modeled by the integral Doi Edwards model, supplemented by chain length fluctuations, have shown very good qualitative agreement with the results obtained by its differential approximation in 4:1:4 constriction domain. Again, much higher Weissenberg numbers can be achieved when the integral model is used. Moreover, very good quantitative results with experimental data of polystyrene solution for the first normal stress difference and shear viscosity defined here as the quotient of a shear stress and a shear rate. Finally, comparison of the two methods used for approximating the time integral constitutive equation, namely Deformation Field Method (DFM) and Backward Lagrangian Particle Method (BLPM), is performed. In BLPM the particle paths are recalculated at every time step of the simulations, what has never been tried before. The results have shown, that in the considered geometries both methods give similar results.